Biomedical subjects
Publications and source records attributed to Raúl O Vallejos.
Using results from the theory of dynamical systems, we derive a general expression for the classical average scattering dwell time tau . Remarkably, tau depends only on a ratio of phase space volumes. We further show that, for a wide class of systems, the average classical dwell time is not in correspondence with the energy average of the quantum Wigner time delay.
The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S(BGS) = -integral dH[P(H)]ln[P(H)], with suitable constraints. Here, we construct and analyze random-matrix ensembles arising from the generalized entropy S(q) = [1- integral dH [P(H)](q)] /(q-1) (thus, S1 = S(BGS) ). The resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form. Making q-->1 recovers the Gaussian ensembles. If q not equal 1, the joint probability distributions P(H) cannot be factorized, i.e., the matrix elements of H are correlated. In the limit of large matrices two different regimes are observed. When q<1, P(H) has compact support, and the fluctuations tend asymptotically to those of the Gaussian ensembles. Anomalies appear for q>1 : Both P(H) and the marginal distributions P( H(ij) ) show power-law tails. Numerical analyses reveal that the nearest-neighbor spacing distribution is also long-tailed (not Wigner-Dyson) and, after proper scaling, very close to the result for the 2 x 2 case--a generalization of Wigner's surmise. We discuss connections of these "nonextensive" ensembles with other non-Gaussian ones, such as the so-called Lévy ensembles and those arising from soft confinement.
The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions. Part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions, because this permits the calculation of correlation times as thermal averages.
The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N-particle Hamiltonian system with a smooth Hamiltonian of the type p(2)+V(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution. In this way, a stochastic process V(t) is defined, and the evolution equation for tangent vectors can now be seen as a stochastic differential equation. An equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cumulant expansion. We show that if cumulants higher than the second one are discarded, the Lyapunov exponent can be obtained by diagonalizing a small-dimension matrix that in some cases can be as small as 3 x 3. In all cases, the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process. We discuss the connection between our approach and an alternative theory, the so-called geometric method.
We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long-range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha>d whereas it decays as an inverse power law of N for alpha<d. A recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results. Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.